It's not at all surprising that this close to sailing both traditional dining times would be booked solid.
But, anytime dining should still allow you to eat within a reasonable amount of time that you require with your meds. On HAL I'd bet half the ship is on meds they need to take with food.
The only real difference at anytime is the need to make a reservation, and the fact you won't have the same tablemates each night.
As Mike suggested you can visit the Maitre D the first day, and see if something opens up at traditional.